Searcharxiv⌕ Search

arXiv subjects

Andrej Dujella

Publications and source records attributed to Andrej Dujella.

47 records · Page 3Linked to original sources
arXiv2012

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion group of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with given torsion group which give the current records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for Z/15Z, there exist an elliptic curve over some quadratic field with this torsion group and with rank >= 2.

math.NT↗
arXiv2011

Ranks of elliptic curves with prescribed torsion over number fields

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call \emph{false complex multiplication}, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.

math.NT↗
arXiv2011

Root separation for irreducible integer polynomials

We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

math.NT↗
arXiv2010

On Hall's conjecture

We show that for any even positive integer d there exist polynomials x and y with integer coefficients such that deg(x) = 2d, deg(y) = 3d and deg(x^3 - y^2) = d + 5.

math.NT↗
arXiv2008

A variant of Wiener's attack on RSA

Wiener's attack is a well-known polynomial-time attack on a RSA cryptosystem with small secret decryption exponent d, which works if d<n^{0.25}, where n=pq is the modulus of the cryptosystem. Namely, in that case, d is the denominator of some convergent p_m/q_m of the continued fraction expansion of e/n, and therefore d can be computed efficiently from the public key (n,e). There are several extensions of Wiener's attack that allow the RSA cryptosystem to be broken when d is a few bits longer than n^{0.25}. They all have the run-time complexity (at least) O(D^2), where d=Dn^{0.25}. Here we propose a new variant of Wiener's attack, which uses results on Diophantine approximations of the form |α- p/q| < c/q^2, and "meet-in-the-middle" variant for testing the candidates (of the form rq_{m+1} + sq_m) for the secret exponent. This decreases the run-time complexity of the attack to O(D log(D)) (with the space complexity O(D)).

cs.CR↗
arXiv2004

Continued fractions and RSA with small secret exponent

Extending the classical Legendre's result, we describe all solutions of the inequality |x - a/b| < c/b^2 in terms of convergents of continued fraction expansion of x. Namely, we show that a/b = (rp_{m+1} +- sp_m) / (rq_{m+1} +- sq_m) for some nonnegative integers m,r,s such that rs < 2c. As an application of this result, we describe a modification of Verheul and van Tilborg variant of Wiener's attack on RSA cryptosystem with small secret exponent.

cs.CR↗
arXiv2000

On the size of Diophantine m-tuples

Let n be a nonzero integer and assume that a set S of positive integers has the property that xy+n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| <= 400 then |S| <= 32, and if |n| > 400 then |S| < 267.81 log|n| (log log|n|)^2. The question whether there exists an absolute bound (independent on n) for |S| still remains open.

math.NT↗
arXiv1998

Diophantine triples and construction of high-rank elliptic curves

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

math.NT↗